### 3.2431 $$\int \frac{1}{x \sqrt{2+4 x-3 x^2}} \, dx$$

Optimal. Leaf size=31 $-\frac{\tanh ^{-1}\left (\frac{\sqrt{2} (x+1)}{\sqrt{-3 x^2+4 x+2}}\right )}{\sqrt{2}}$

[Out]

-(ArcTanh[(Sqrt[2]*(1 + x))/Sqrt[2 + 4*x - 3*x^2]]/Sqrt[2])

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Rubi [A]  time = 0.0109847, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.111, Rules used = {724, 206} $-\frac{\tanh ^{-1}\left (\frac{\sqrt{2} (x+1)}{\sqrt{-3 x^2+4 x+2}}\right )}{\sqrt{2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/(x*Sqrt[2 + 4*x - 3*x^2]),x]

[Out]

-(ArcTanh[(Sqrt[2]*(1 + x))/Sqrt[2 + 4*x - 3*x^2]]/Sqrt[2])

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{x \sqrt{2+4 x-3 x^2}} \, dx &=-\left (2 \operatorname{Subst}\left (\int \frac{1}{8-x^2} \, dx,x,\frac{4+4 x}{\sqrt{2+4 x-3 x^2}}\right )\right )\\ &=-\frac{\tanh ^{-1}\left (\frac{\sqrt{2} (1+x)}{\sqrt{2+4 x-3 x^2}}\right )}{\sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.0076599, size = 28, normalized size = 0.9 $-\frac{\tanh ^{-1}\left (\frac{x+1}{\sqrt{-\frac{3 x^2}{2}+2 x+1}}\right )}{\sqrt{2}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/(x*Sqrt[2 + 4*x - 3*x^2]),x]

[Out]

-(ArcTanh[(1 + x)/Sqrt[1 + 2*x - (3*x^2)/2]]/Sqrt[2])

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Maple [A]  time = 0.042, size = 29, normalized size = 0.9 \begin{align*} -{\frac{\sqrt{2}}{2}{\it Artanh} \left ({\frac{ \left ( 4+4\,x \right ) \sqrt{2}}{4}{\frac{1}{\sqrt{-3\,{x}^{2}+4\,x+2}}}} \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(-3*x^2+4*x+2)^(1/2),x)

[Out]

-1/2*2^(1/2)*arctanh(1/4*(4+4*x)*2^(1/2)/(-3*x^2+4*x+2)^(1/2))

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Maxima [A]  time = 1.49362, size = 47, normalized size = 1.52 \begin{align*} -\frac{1}{2} \, \sqrt{2} \log \left (\frac{2 \, \sqrt{2} \sqrt{-3 \, x^{2} + 4 \, x + 2}}{{\left | x \right |}} + \frac{4}{{\left | x \right |}} + 4\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(-3*x^2+4*x+2)^(1/2),x, algorithm="maxima")

[Out]

-1/2*sqrt(2)*log(2*sqrt(2)*sqrt(-3*x^2 + 4*x + 2)/abs(x) + 4/abs(x) + 4)

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Fricas [A]  time = 2.06355, size = 111, normalized size = 3.58 \begin{align*} \frac{1}{4} \, \sqrt{2} \log \left (-\frac{2 \, \sqrt{2} \sqrt{-3 \, x^{2} + 4 \, x + 2}{\left (x + 1\right )} + x^{2} - 8 \, x - 4}{x^{2}}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(-3*x^2+4*x+2)^(1/2),x, algorithm="fricas")

[Out]

1/4*sqrt(2)*log(-(2*sqrt(2)*sqrt(-3*x^2 + 4*x + 2)*(x + 1) + x^2 - 8*x - 4)/x^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \sqrt{- 3 x^{2} + 4 x + 2}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(-3*x**2+4*x+2)**(1/2),x)

[Out]

Integral(1/(x*sqrt(-3*x**2 + 4*x + 2)), x)

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Giac [B]  time = 1.26917, size = 132, normalized size = 4.26 \begin{align*} -\frac{1}{6} \, \sqrt{6} \sqrt{3} \log \left (\frac{{\left | -14 \, \sqrt{10} - 14 \, \sqrt{6} + \frac{28 \,{\left (\sqrt{3} \sqrt{-3 \, x^{2} + 4 \, x + 2} - \sqrt{10}\right )}}{3 \, x - 2} \right |}}{{\left | -14 \, \sqrt{10} + 14 \, \sqrt{6} + \frac{28 \,{\left (\sqrt{3} \sqrt{-3 \, x^{2} + 4 \, x + 2} - \sqrt{10}\right )}}{3 \, x - 2} \right |}}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(-3*x^2+4*x+2)^(1/2),x, algorithm="giac")

[Out]

-1/6*sqrt(6)*sqrt(3)*log(abs(-14*sqrt(10) - 14*sqrt(6) + 28*(sqrt(3)*sqrt(-3*x^2 + 4*x + 2) - sqrt(10))/(3*x -
2))/abs(-14*sqrt(10) + 14*sqrt(6) + 28*(sqrt(3)*sqrt(-3*x^2 + 4*x + 2) - sqrt(10))/(3*x - 2)))